Only you can decide if they would make good babysitters. Perhaps you might consider your kids only being with the relatives while you are with them. That way you can prevent any indoctrination. It does make a difference how old your children are when it comes to exposure to things you deem negative. Also, remember your kids will get exposed to a lot of different viewpoints in school and in life and you will have to navigate those as well. How you teach them to deal with diversity and differences in opinions and beliefs will shape how they will do the same in the future.
If they enjoy Christmas then you could explain how some people (like Jews, Muslims, Hindus and Buddhists) don't celebrate it but that is no reason for them not to enjoy it. They are under your care until they are adults but they are their own individuals and will make their own choices when they get older. All you can do is give the best guidance and all the love and hope for the best.